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Robust Small-Signal Detection

Module by: Don Johnson

These results can be modified to derive a robust detector that accommodates small signals. To make this detector insensitive to deviations from the nominal noise probability density, the density should be replaced by the worst-case density consistent with the maximum allowed deviations. For unimodal, symmetric densities, the worst-case density equals that of the nominal within a region centered about the origin while being proportional to an exponential for values outside that region. p ω nn=1-ε p o nnif|n|< n ′ 1-ε p o n n ′ ⅇ-a|n|- n ′ if|n|> n ′ p ω n n 1 ε p o n n n n ′ 1 ε p o n n ′ a n n ′ n n ′ The parameter aa controls the rate of decrease of the exponential tail; it is specifies by requiring that the derivative of the density's logarithm be continuous at n= n ′ n n ′ . Thus, a=-ddnln p o nn a n p o n n for n= n ′ n n ′ . For the worst-case density to be a probability density, the boundary value n ′ n ′ and the deviation parameter εε must be such that its integral is unity. As εε is fixed, this requirement reduces to a specification of n ′ n ′ . ∫- n ′ n ′ p o nndn+2a p o n n ′ =11-ε n n ′ n ′ p o n n 2 a p o n n ′ 1 1 ε For the case where the nominal density a= n ′ σ2 a n ′ σ 2 and this equation becomes 1-2Q n ′ σ+σ n ′ 2πⅇ- n ′ 22σ2+11-ε 1 2 Q n ′ σ σ n ′ 2 n ′ 2 2 σ 2 1 1 ε The non-linearity required by the small-signal detection strategy thus equals the negative derivative of the logarithm of the worst-case density within the boundaries while equaling a constant outside that region.

frl=-ddnln p o nn|n=rlif|rl|< n ′ asignrlif|rl|> n ′ f r l n r l n p o n n r l n ′ a sign r l r l n ′ (1)
The robust detector for small signals thus consists of a clipper followed by a matched filter for each signal, the results of which are compared with a threshold.

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